Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,7,12)(3,10,5,8)(4,14,13,9)(6,11,15,16)(19,25)(22,29)(26,31)(30,32);(1,3,7,5)(2,8,12,10)(4,15,13,6)(9,16,14,11)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(1,4)(2,9)(3,6)(5,15)(7,13)(8,11)(10,16)(12,14)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(1,5,7,3)(2,10,12,8)(4,15,13,6)(9,16,14,11);(1,6,7,15)(2,11,12,16)(3,13,5,4)(8,14,10,9)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32);(1,2)(3,8)(4,9)(5,10)(6,11)(7,12)(13,14)(15,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #185428, #185434, #185440, … |
| maximal normal subgroup of | #185428, #185434, #185440, … |
| maximal quotient of | #7626234, #7626240, #7626274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55825);